A proof-theoretical investigation of Zantema's problem

Abstract

We present a concrete example of how one can extract constructive content from a non-constructive proof. The proof investigated is a termination proof of the string-rewriting system 1100 → 000111. This rewriting system is self-embedding, so the standard termination techniques which rely on Kruskal's Tree Theorem cannot be applied directly. Dershowitz and Hoot [3] have given a classical termination proof using a minimal bad sequence argument. We analyse their proof and give a constructive interpretation of it, which enables us to extract a first proof in Type Theory that uses generalised inductive definitions. By simplifying this constructive proof we obtain a second proof in a theory conservative over primitive recursive arithmetic. This proof is generalised to a theorem about string rewriting systems.KeywordsOpen InductionProof TheoryConstructive ProofTermination ProofInductive DefinitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.